| L(s) = 1 | + 11.5i·3-s − 231.·7-s + 108.·9-s + 559. i·11-s − 107. i·13-s + 441.·17-s + 1.87e3i·19-s − 2.68e3i·21-s − 3.83e3·23-s + 4.07e3i·27-s + 3.36e3i·29-s + 7.95e3·31-s − 6.48e3·33-s + 1.06e4i·37-s + 1.25e3·39-s + ⋯ |
| L(s) = 1 | + 0.743i·3-s − 1.78·7-s + 0.446·9-s + 1.39i·11-s − 0.177i·13-s + 0.370·17-s + 1.19i·19-s − 1.32i·21-s − 1.51·23-s + 1.07i·27-s + 0.744i·29-s + 1.48·31-s − 1.03·33-s + 1.28i·37-s + 0.131·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.5464277134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5464277134\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 11.5iT - 243T^{2} \) |
| 7 | \( 1 + 231.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 559. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 107. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 441.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.87e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 3.83e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.36e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 7.95e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.06e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 9.96e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 925. iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 8.06e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 7.95e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 1.68e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 1.12e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 3.36e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 8.86e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.55e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.94e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.02e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 9.24e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.86e4T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.965327287991973077483317782101, −9.732486282206262698381751056529, −8.519702580257030735517287136212, −7.39165805646151480008715083274, −6.61900138018782737815354564497, −5.75703300798018064365204413912, −4.54211727578384964109809545914, −3.79943443174194161713384194470, −2.89262252142159593786964731452, −1.50972909546318885739284648934,
0.14084815648404601439808377300, 0.821467616039885812062343391732, 2.34487496336961606673315640316, 3.25752862759277821369152563139, 4.20516681787457169756255757504, 5.79535493991217841775337360206, 6.34840419425341870472603459783, 7.05068201299960722068627753270, 8.027474014460970399086944292357, 8.955392916343921477121183495162
